Dynamical System (definition) - Measure Theoretical Definition

Measure Theoretical Definition

See main article measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the triplet (T, (X, Σ, μ), Φ) Here, T is a monoid (usually the non-negative integers), X is a set, and Σ is a topology on X, so that (X, Σ) is a σ-algebra. For every element σ in Σ, μ is its finite measure, so that the triplet (X, Σ, μ) is a probability space. A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ−1(σ) ∈ Σ. A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has μ(Φ−1(σ)) = μ(σ). Combining the above, a map Φ is said to be a measure-preserving transformation of X, if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), φ), for such a Φ, is then defined to be a dynamical system.

The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be finite time evolution map and the construction is more complicated.